III.
Prof. J.
C.
SOLID STATE PHYSICS
Slater
Prof. G. G. Harvey
Prof. A. F.
Prof. L.
Dr. R.
J.
Dr. R.
Malvano
L. Jennings
R. H. Kingston
A. Meckler
Dr. E. H. Sondheimer
Kip
C.
Tisza
F.
D. Reiner
Davis
E. M.
A.
Harrison
Gyorgy
A SIMPLIFIED HARTREE METHOD FOR ENERGY BANDS
Since the early days of the quantum-mechanical treatment of molecules and solids,
two types of approximate calculation of solutions of the Schrodinger equation have been
used with varying success.
The first of these is the Heitler-London method.
It assumes
that in a molecule or solid we can use the wave function which an electron would have in
an isolated atom as a first approximation, proceeding from there by perturbation theory.
It is at the basis of a good deal of the theory of molecular valence,
theory of ferromagnetism,
and of the theory of paramagnetism.
that of molecular orbitals, when applied to molecules,
to metals.
of Heisenberg's
The other theory is
or of energy bands,
when applied
It starts by using wave functions which are solutions of a many-center
problem (for molecules) or of a periodic potential problem (for metals and crystals).
Both methods have their. advantages and disadvantages.
Heitler-London Method
Advantages:
It is easy to understand and describe; leads to exchange integrals con-
cerned in interatomic forces and magnetic problems.
excessive charge fluctuations,
Wave functions are easily set up;
the piling up of too much charge on an atom,
are
eliminated.
Disadvantages:
Wave functions are not orthogonal; there is great complication when
quantitative calculations are attempted.
wrong sign.
Exchange integrals for ferromagnetism are of
No real explanation of ferromagnetism is possible.
The method fails to
account for energy bands in metals, energy gap theory.of semiconductors, properties
of insulators.
Molecular Orbital Method
Advantages:
Wave functions are orthogonal.
lators follows directly.
Energy ban theory of metals and insu-
Application to ferromagnetism leads to correct results; predicts
which elements should be ferromagnetic.
Provides a wide application to study of
chemical valence and band spectra.
Disadvantages:
Leads to excessive charge fluctuations,
and as usually used, leads
to wrong limiting behavior at large internuclear distances.
Examination of these advantages and disadvantages will probably convince the reader
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SOLID STATE PHYSICS)
that the molecular orbital, or energy band, method is the best approximation to use.
The great advance in semiconductor theory since the war justifies this view, leading to
very precise ways of measuring gap widths between energy bands. The writer therefore
feels that the most important step toward further development of the electronic theory
of solids depends on putting the energy band or molecular orbital method on the soundest
basis possible, and then using it in a straightforward way for making all sorts of calculations. Many problems being worked on experimentally in the various laboratories at
M.I. T. have theories which could be improved by better study of this method.
We may
enumerate the following:
1.
Soft x-ray emission spectra of solids (solid-state group, Research Laboratory
of Electronics), which depends very directly on calculation of energy bands.
2.
Semiconductor work (physical electronics group, Research Laboratory of Elec-
tronics, and Insulation Laboratory), in which the width of the energy gap can be measured fairly accurately, but calculations so far fail completely to reproduce the experimental values.
The Insulation Laboratory is interested not only in gap width, but also
in its change with pressure, which should give particularly valuable information to check
the theory.
3.
Paramagnetism of iron group and rare earth salts (solid-state group, Research
Laboratory of Electronics, low temperature group, Research Laboratory of Electronics,
for application to adiabatic demagnetization, and Insulation Laboratory). These problems have generally been treated by the Heitler-London method; it is very desirable to
fit them into the framework of the energy band theory, and also to tie in the ground state
of the paramagnetic ions, which alone is concerned in paramagnetism, with the excited
levels and spectroscopic behavior of the salts.
4.
Ferromagnetism of metals and oxides (solid-state group, Research Laboratory
of Electronics, and Insulation Laboratory). The ferromagnetic oxides, such as Fe O04
3
and the ferrites, being studied by the Insulation Laboratory, are of particularly great
interest here, for they combine ferromagnetic and antiferromagnetic behavior, and
further have X-point transitions below which the electrons are ordered (some of the ions
are Fe + + and some Fe + + + ) in a regular arrangement, while above the transition they are
disordered.
Below the transition the properties could be described alternatively by the
Heitler-London or the energy band method, while above it the energy band method is
practically necessary.
Solution of this problem will almost certainly lead to increased
insight into the relations of the two methods.
5.
Superconductivity (solid-state group, Research Laboratory of Electronics).
There is a close relation between the ordering of electrons below the X-point in the ferromagnetic oxides, and in Tisza's theory of superconductivity.
The same techniques of
explaining the relations between energy band and Heitler-London methods which must be
worked out for the ferromagnetic oxides would presumably be useful in that theory of
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(III.
SOLID STATE PHYSICS)
superconductivity as well, if it proves to be correct.
6.
ment).
Order-disorder in alloys (x-ray group, physics department; metallurgy departThe energy band theory meets this problem in several aspects.
First, the
energy levels are distorted in an alloy, and a study of this distortion, which has not been
made,
should throw much light on the problems of solid solutions.
In addition, new
work at Oak Ridge with neutron diffraction shows that antiferromagnetic substances have
an order-disorder situation for magnetic ions which bears close analogy to orderdisorder in alloys,
7.
and which ties in closely with energy band theory.
Chemical valence (chemistry department).
ment of the quantum theory of valence,
Physicists were active in the develop-
but have avoided it in recent years.
It has
developed far in a qualitative way, but the quantitative work has been mostly very crude,
from the point of view of a physicist. A reliable quantitative theory of molecular orbitals is practically essential to a further development of valence theory.
With all of these problems in mind, the writer is starting a program of reexamining
the molecular orbital method.
The first step in this program is to find the best poten-
tial function to use for our one-electron problem.
extension of Hartree's self-consistent field method.
The molecular orbital method is an
In that method we replace the
actual force acting on an electron, which comes from the nucleus and all other electrons,
and hence depends on the position of all other electrons, by an averaged potential of
some sort, depending only on position.
This is entirely analogous to the method of
handling space charge in an electronic problem.
The averaged potential is the electrostatic potential of the nuclei, and of the average charge density of all electrons except
the one in question. If there are n electrons in the problem, we can consider the charge
density of the electrons to be the density of all n electrons, minus a correction charge
density for one electron, arising because the electron in question does not act on itself.
Once we have set up this potential, we find solutions of Schr6dinger's equation in it, get
from these solutions the charge density of each electron, add these to get the total charge
density, and demand that that agree with the charge density initially assumed. This is
the requirement of self-consistency, which is characteristic of Hartree's method.
All this is clear and straightforward,
of the correction charge density.
and the only question comes in the formulation
Two methods have been proposed for this, neither
one wholly satisfactory.
Hartree proposed that the correction charge density depend on
which electron of the problem we were considering, and that it be just the charge density
of that electron itself; that is,
that the electron be acted on by all other electrons, each
distributed according to its one-electron wave function.
For an isolated atom,
the
It seems at first sight intuitively obvious, until we remember
that on account of the antisymmetry of the wave function we cannot distinguish between
method works fairly well.
different electrons, so that it is hardly possible uniquely to say that an electron has a
given one-electron function. We can see the difficulty more clearly if we consider the
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(IiI.
SOLID STATE PHYSICS)
energy band problem.
In an atom, the Hartree method removes the correction charge
from the atom itself, so that the electron is acted on by the nucleus and (n - 1) other
electrons.
But in a crystal, the wave function of an electron is a wave corresponding
to equal amplitudes on each of the N atoms of the crystal, so that the correction charge
would be equally divided between N atoms, giving only 1/Nth of an electronic charge on
each, which could be neglected for large N.
quite wrong.
Hence for a crystal the Hartree method is
Another disadvantage of the Hartree method is that each electron moves
in a different field, so that the wave functions of different electrons are not orthogonal.
The Hartree-Fock method avoids some of these difficulties.
It starts by assuming
an antisymmetric wave function, and hence automatically takes account of the identity
of electrons. Like the Hartree method, its potential is different for each electron, and
the correction charge density is determined in a complicated way involving exchange
integrals between electrons.
There are two great advantages over the Hartree method,
particularly for molecular orbital and energy band work. First, the exchange terms
are such that the correction charge is removed from the general neighborhood of the
point where the electron is located.
Thus in a crystal of N atoms, if the electron happens
to be on one of the atoms, the correction charge is largely located on that atom; it moves
around as the electron moves.
havior.
Therefore the method has the qualitatively correct be-
Second, the wave functions of the various electrons are orthogonal.
Counter-
balancing these advantages is the very serious practical disadvantage that the method
is so hard to apply that it is almost impossible in practice to work it out for a molecule
or solid; even for a single atom, it has been applied only to the lighter atoms (Cu
+
is the
heaviest one), and it gets almost impossibly hard for a heavy atom.
There is one case in which the Hartree-Fock method becomes very simple: the perfect gas of free electrons.
If we have a volume filled with free electrons, with a uni-
form positive charge density just sufficient to balance the average negative charge of
the electrons, the Hartree-Fock equations become simply the equations of a free particle,
which can be immediately solved. The electrons obey the Fermi exclusion principle,
and form a model for the Sommerfeld free-electron picture of a metal. The correction
charge then proves to be a density, depending on the distance from the electron in
question, and on its momentum, equal to the total density of electrons of the same spin
as the electron in question at the location of the electron, and rapidly falling off with
distance.
If the charge density of electrons is p, we can say in a very crude way that the
4
3
correction charge density is approximately p within a sphere of radius r, where -3 7r p
equals the electronic charge, and is zero outside that sphere. We can readily find the
correction to the potential on account of the correction charge:
dimensionally it depends
on the charge e, divided by the radius r of the sphere, or is proportional to pl/3.
This
correction depends somewhat on whether the electron is at the bottom or top of the Fermi
distribution. If it is at the bottom the correction is numerically greater, decreasing as
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(III.
SOLID STATE PHYSICS)
we go to the top, and for an excited electron, well above the Fermi level, the correction
goes to zero, since such an excited electron has a momentum quite different from any
of the Fermi electrons, and the exclusion principle does not push the electrons out of its
way very effectively, so that the correction charge is much more diffuse, and its potential correction is numerically smaller.
One can then set up a correction to the potential, using this free-electron calculation,
1/3 This correction should depend on the energy or momenwhich is proportional to p
tum of the electron; but we can average over the energy, and get a unique correction,
1/3 The writer is now exploring the possibility of using this correction
proportional to p
in the case where we have a potential varying with position, as in the Hartree method.
It is well known that the Fermi-Thomas method of treating atomic structure handles the
electrons at each point of a potential field as if they were free electrons.
where the potential is V, so that the potential energy of an electron is -eV,
At a point
we assume
that there are energy levels as for a free electron gas of Fermi energy EF, in a region
of uniform potential energy -eV,
value -eV up to E F
.
so that the energy levels are filled all the way from the
This leads, according to Fermi statistics, to a definite charge den-
sity at each point of space, and we then apply Poisson's equation to derive a potential
from this charge density, and make the self-consistent demand that this agree with the
potential originally assumed.
The Fermi-Thomas method neglects completely the wave
nature of the electronic motion, and the electron shells; thus it does not lead to the
periodic system of the elements.
Nevertheless it leads to fairly accurate values for the
potential at points inside an atom or molecule or crystal.
The Fermi-Thomas method as ordinarily used assumes that an electron acts on itself;
it neglects the correction charge which we have been so far assuming.
It has been modi-
fied, however, by using the correction to the potential proportional to pl/3 , which we
have described in the preceding paragraphs. This improves the Fermi-Thomas method
considerably, but it retains its fault of not considering properly the various electron
shells. The present suggestion is then a simple one: to use the correction to the potential given by p
, as described in the free-electron picture, along with the Hartree
method. Here, in an atom with n electrons, we would be treating the potential of the n
electrons properly; the free-electron correction would come in only in the correction
charge, which is 1/nth of the total electronic charge.
The wave functions of the elec-
trons would be found in this corrected potential by Schrodinger's equation, so that we
should take proper account of the shell structure. The result, then, would be close to
that of the Hartree-Fock method, though not quite so accurate, and would be very much
better than the Fermi-Thomas method, with or without exchange. There would be a
unique potential function, and hence the wave functions would be orthogonal. Finally, the
method would be very simple to apply, even simpler than the elementary Hartree method,
since one potential would be used for all electrons, and the correction proportional to
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(III.
SOLID STATE PHYSICS)
pl/3 is extremely easy to set up. The potential qualitatively is correct in the energy
band problem of the crystal, as the Hartree-Fock method is; it avoids the shortcomings
of the Hartree method.
A check of the method is at present being made by the writer and Mr. George W.
Pratt, Jr., on the atomic problem of Cu+, which has been handled by the Hartree and
Hartree-Fock methods. Preliminary indications are that the results will lie intermediate between those of the Hartree and Hartree-Fock methods, or near the HartreeFock case, the deviations being somewhat different for the wave functions of the various
electrons. The method gives promise of being so much simpler than the Hartree-Fock
method that it should be practicable to handle even the heavy atoms by it. And it should
provide a direct and straightforward approach to finding the self-consistent field for
determining molecular orbitals and energy bands for molecules and solids.
J. C. Slater
B.
HARTREE CALCULATIONS FOR ATOMS
The differential analyzer is being used for calculation of the wave functions of the
Cr+ ion, using the Hartree method without exchange. This ion was chosen as a simple
one, near others which had already been computed, and of considerable interest on
account of its position midway through the iron group; it was done largely for practice,
to see how great the problem of calculation of atomic wave functions is if the differential
analyzer is available.
Calculations are almost completed.
The differential analyzer
is also being used for a check of the simplified treatment of exchange mentioned elseThis simplified Hartree method is being applied to the Cu + ion,
the heaviest which has been solved by both the Hartree and Hartree-Fock methods, to
compare the results of the three methods. This work is still in its early stages.
J. C. Slater, G. W. Pratt, Jr.
where in this report.
C.
QUANTUM THEORY OF ANTIFERROMAGNETISM
The quantum-mechanical description of the lowest state of an antiferromagnetic
substance reflects the non-classical nature of a spin system, which prevents attainment of simply ordered structures.
The situation is analagous to the chemical problem
and there exists the phenomenon of "resonance" among different bond structures such
as occurs in certain molecules. Thus the concept of "bond order" which has been
applied to the molecular case is also a useful concept to apply to antiferromagnetism
since it is a measure of the correlation in direction of the spin magnetic moments of
neighboring electrons.
The eigenfunction describing the spin state of the antiferromagnetic system may be
expanded in terms of a set of independent spin functions. Two such sets have been used
frequently.
One of these we call the "spin product-functions. " These functions are
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SOLID STATE PHYSICS)
(III.
eigenfunctions of S z but not of S .
For this reason, the secular equation to which they
lead is of higher order than that which results from use of the "valence bond functions"
which are eigenfunctions of both S 2 and S z
.
A formal solution to the problem of a linear chain of spins has been given by Bethe.
The energy in the limiting case of very large numbers of spins is given simply; but one
cannot tell very much about the eigenfunction corresponding to this solution.
Another approach to the problem of the infinite linear chain is the approximation
developed by Slater and byHulthen, in which the eigenstate is expanded in terms of
spin product-functions,
with some simplified assumptions for the coefficients.
A different approximation, based on the valence bond functions,
molecular problems.
has been used in
The assumption is made that all canonical structures having the
same degree of excitation, i. e.
same number of "double bonds, " enter in the correct
eigenfunction with the same coefficient.
utilize this approximation.
We have modified the Slater-Hulthen method to
One would expect to obtain an improved result since the
expansion is carried out in terms of functions having the same multiplicity as the correct
eigenfunction;
and we do in fact obtain a value for the energy which is closer to Bethe's
"exact" value than are the values obtained by Slater and Hulthen.
The canonical struc-
tures which are found to contribute appreciably to the eigenfunction are all within an
extremely narrow range of the degree of excitation.
This work is described in more detail in a doctoral thesis in physics submitted by
R. J.
Harrison, "The Quantum Theory of Antiferromagnetism. "
L.
D.
Tisza, R. J.
Harrison
SOFT X-RAY VACUUM SPECTROGRAPH
An evaporating system has been designed and constructed.
It consists of a small
tantalum cup about 3/16 inch in diameter and 5/16 inch long, easily removable for refilling or replacement,
placed inside a helical tungsten filament and heated by radiation,
and electron bombardment with a power of 50 to 75 watts maximum.
0
possible to attain temperatures up to about 2000 C.
aluminium,
This makes it
The unit has been tested with
in an auxiliary glass vacuum system in which the progress of the operation
can be observed.
A uniform coating covering an area greater than that needed for the
x-ray target was easily attained, and the evaporation unit is now considered satisfactory.
Further trial runs of the spectrograph with x-rays have been postponed in order to
improve the optical alignment,
and investigate the source of extraneous scattering giving
rise to a high background intensity.
focusing of a line spectrum,
Since the ultimate test of proper alignment is sharp
modifications were made toward this end.
A high vacuum
spark source operating at about 50 kv was built into the x-ray chamber, with the electrodes located where the x-ray target is normally.
Either solid aluminium electrodes,
or hollow aluminium charged with sodium hydroxide were used.
-23-
Ilford Q2 plates were
(III.
SOLID STATE PHYSICS)
used and since they cannot be readily bent to a radius as small as one meter (the radius
of the Rowland circle) without breaking, only a small region of the spectrum could be
explored at one time.
Two tentatively identified lines on one plate and one on another
are about as sharp as the slit width would permit and indicate a reasonabry good alignment of the grating system.
Electrical interference caused by the 50-kv spark was so excessive, even with extensive shielding, that it was not possible to use the photomultiplier circuits with the
spark source.
We have recently secured some 35-mm Eastman Type SWR film which has properties
similar to the Ilford plates in the extreme ultraviolet, and intend to study the line spectrum further with it.
The extraneous scattering has been reduced somewhat by means
of guard slits and baffles, but is still too high.
G. G. Harvey, E. M. Gyorgy, R. H. Kingston, A. Meckler
E.
PARAMAGNETIC RESONANCE EXPERIMENTS
During the past four months primary emphasis has been given to improvement of
measuring techniques and apparatus. A number of improvements have been made in the
sensitivity and stability of our apparatus, and in addition we are now able to make
routine measurements down to temperatures of 2'K. We are just starting to take advantage of our improved equipment and have already found a number of extremely interesting
results in the several kinds of single crystals which we have had under observation.
Since the improvement in our technique has played a major role in our ability to obtain
our interesting results, we shall discuss briefly the important details of our present
equipment.
And we shall describe the most interesting of our preliminary results on
the several salts tested.
The low temperature helium cryostat is of simple and conventional design.
The
outer glass dewar which holds liquid nitrogen is evacuated, and the inner dewar, containing the liquid helium and microwave cavity, is sealed off with 1-mm pressure of
nitrogen at room temperature.
filling with helium.
This is the usual technique for allowing precooling before
The inside of the inner dewar is made just large enough to allow
insertion of standard 1-inch by 1/2-inch 3-cm wave guide.
The thickness of the brass
wave guide has been reduced from its original 0. 050 inch to about 0. 015 inch to
reduce heat losses and heat capacity.
The cavity is a one-wave-length piece of guide.
Small holes are drilled in both guide and cavity to allow liquid helium inside.
This in-
creases the helium capacity of the cryostat and in no way interferes with the microwave
resonance measurements.
The coupling between cavity and guide can be changed by
inserting irises with various hole sizes between cavity and guide.
It is interesting to note that 4 0 -temperatures can be maintained in the cryostat for
as long as four hours with a single filling of about 3 liters of liquid helium, much of
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(III.
SOLID STATE PHYSICS)
which is used up in the initial cooling.
In common with others using pyrex glass cryostats, we find that after about six runs with liquid helium the heat leak through the inner
dewar increases markedly and it is necessary to re-pump the inner dewar, presumably
to remove helium gas which has leaked in.
It is possible to evacuate the entire contents
of the inner dewar, including the cavity and wave guide.
Before precooling, the system
is evacuated and filled with helium gas to allow heat transfer.
The cryostat is 53 cm
long, and narrows down to 6. 4-cm diameter at the bottom to allow insertion between the
poles of an electromagnet.
The method of measurement is similar to that described in previous Progress Reports (see also Research Laboratory of Electronics Technical Reports No. 91 and
No. 61).
Variations in the power absorbed by the crystal as a function of the externally
applied magnetic field is measured by means of the change in the power reflected by the
cavity.
This change is a consequence of the change in the Q of the cavity containing the
crystal. The modifications which have been made are primarily in the direction of
improving stability and sensitivity.
A major improvement in stability has been obtained
by changing the method of amplitude modulation of the microwave power. We now run
the klystron at a constant frequency, and obtain amplitude modulation by inserting a
crystal in a tee in the transmission wave guide. Modulating voltage placed across the
crystal changes its impedance and hence modulates the amplitude of the signal to the
cavity.
This rather standard modulation technique has a tremendous advantage, since
it allows the klystron to be operated with batteries and hence eliminates the rather
serious noise on the a-c lines and the frequency drift due to a-c line variation. A Pound
stabilizer would give the necessary frequency stability without batteries but it is not
certain that all of the line noise could be eliminated. In any case, power requirements
are small and therefore battery operation presents no difficulties. With the increased
stability of the power source, it became useful to improve the stability of the amplifier
used on the audio frequency signal resulting from rectifying the microwave signal. At
present we are operating this amplifier with batteries also, and have considerably improved the signal-to-noise ratio. We have not reached the ultimate possible sensitivity,
but much further increase will bring us to the point where slight mechanical instability
of the wave guide will begin to limit us. An increase of a factor of 2 to 4 is probably
available to us by improving the Q of our cavity, and our frequency stability would
justify such an improvement.
However, our present sensitivity has opened up a considerable field of investigation, and we will pursue this for the present, making im-
provements as convenient.
We have another type of equipment which is an improved version of equipment built
by Malvano (1) and used by him in Italy prior to his visit here. In this apparatus the
external magnetic field can be modulated at 60 cps and the resulting absorption curve can
be presented directly on a CRO.
This method is exceedingly useful, especially as a
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(III.
SOLID STATE PHYSICS)
supplement to our original method.
Actually the methods we are using are fairly conventional microwave techniques which
have been applied in many allied fields. It is rather surprising, when there is so much
to be gained, that they have not been used before in the paramagnetic resonance field.
We will now report briefly on the results we have so far obtained. During the
process of improving our equipment and testing the operation of the cryostat we used a
single crystal of (NH 4 ) Cr(S0 4 ) 2 - 12 H 2 0, which has been measured by several others
(1, 2, 3, 4), for testing our apparatus. Actually we had a special motive for using this
salt, since Malvano had found the faint suggestion of an extra absorption peak at
low magnetic fields. We hoped that our new apparatus would have sufficient sensitivity
to determine whether this peak was real or whether it had to do with instrumental effects.
Operation with both of our types of equipment have shown the peak to be real, and we have
now made runs at five temperatures from room temperature down to 4°K. Complete
results will be published soon; we can say this much at present: The presence of the
extra peak indicates the existence of a double Stark splitting of the ground level of the
paramagnetic Cr+++ ion at all temperatures. Data of previous workers (4, 5) indicated
0
double splitting only below 90 K. We have further shown that the double splitting cannot
be explained by assuming that one or more of the ions in the unit cell is different from
the others.
In order to clarify some of the discordant experimental data on some paramagnetic
rare earth salts we have undertaken to measure a number of them at the low temperatures
required. In this work we are collaborating with Prof. F. H. Spedding and Mr. Buell
Ayers who have provided the single crystals which we are using. So far, preliminary
measurements have been made on three crystals, and the most interesting results will
be briefly described.
Pr C13- 6 H20 gives a narrow resonance at 4'K at about 2300 gauss for a particular
orientation of the crystal with respect to the external magnetic field. The width of the
resonance is about 50 gauss at the half-power point. There is evidence of resolved
hyperfine structure in this peak.
The most interesting of the rare earth salts which we have tested is Nd 2 (SO 4 ) 3
8H 20. As has been previously pointed out, this crystal presents large anisotropy at
low temperatures. In the easy direction of magnetization we find an anomalous disperThe frequency of the resonant cavity containing the single crystal changes
rapidly (and possibly discontinuously) at a certain applied magnetic field. The magnetic
field necessary to cause this transition decreases with temperature. Further work on
sion curve.
this interesting effect is now in progress and will be reported in detail later.
A. F. Kip, R. Malvano, C. F. Davis, L. Jennings, D. Reiner
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(III.
SOLID STATE PHYSICS)
References
1.
R. Malvano, M. Panetti:
2.
P.
Weiss:
3.
C.
A. Whitmer, et al: Phys.
4.
B. Bleaney, R. P.
5.
B. Bleaney:
F.
THE INFLUENCE OF A TRANSVERSE MAGNETIC FIELD ON THE
Phys. Rev. 73,
Nuovo Cimento 7, 28 (1950).
470 (1948).
Penrose:
Phys. Rev. 75,
Rev. 74,
1478 (1948).
Proc. Phys. Soc. 60,
395 (1948).
1962 (1949).
CONDUCTIVITY OF THIN METALLIC FILMS (1)
The numerical evaluation of the theory for the case of partially elastic surface
scattering has now been completed by the Joint Computing Group.
the theory will be published as Technical Report No.
Complete details of
161 by this Laboratory.
E. H. Sondheimer
Reference
1.
See Proceedings of the M.I.T. International Conference on the Physics of Very Low
Temperatures,
p.
M.I.T. January 15,
105; Quarterly Progress Report, Research Laboratory of Electronics,
1950, p. 23; Nature 164, 920 (1949).
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